Which Method Correctly Computes the Fourier Transform of the Ramp Function?

Question

While calculating the Fourier transform of the ramp function, I found two different methods that yield different results. Here is how each method works before I share my confusion.

Define the ramp function as: $$ R(x) = \begin{cases} 0, & x < 0 \\ x, & x \geq 0 \end{cases}\tag{1} $$ We start with the Fourier transform defined as (with normalization constant $\frac{1}{\sqrt{2\pi}}$): $$ f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \tilde{f}(k) e^{ikx} \, dk, \quad \tilde{f}(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-ikx} \, dx.\tag{2} $$ Method 1: First, compute the Fourier transform of the constant function $h(x) = \frac{1}{2}$, and then of the sign function, where: $$ \mathrm{sgn}(x) = \begin{cases} -1, & x < 0 \\ 1, & x \geq 0 \end{cases}\tag{3} $$ Using linearity, the Fourier transform of the step function (Heaviside function) $H(x) = h(x) + \frac{1}{2}\mathrm{sgn}(x)$ is: $$ \tilde{H}(k) = \mathcal{F}\left[h(x) + \frac{1}{2}\mathrm{sgn}(x)\right] = \mathcal{F}[h(x)] + \frac{1}{2}\mathcal{F}[\mathrm{sgn}(x)] = \frac{\sqrt{2\pi}}{2}\delta(k) - \frac{i}{\sqrt{2\pi}k},\tag{4} $$ Using the property of Fourier transforms for integrals: $$ \mathcal{F}[g(x)] = \frac{1}{ik} \tilde{f}(k), \quad \text{where} \,\, g(x) = \int_{-\infty}^x f(t) \, dt,\tag{5} $$ and noting that: $$ R(x) = \int_{-\infty}^x H(t) \, dt,\tag{7} $$ we get: $$ \mathcal{F}[R(x)] = \frac{1}{ik} \tilde{H}(k) = -\frac{1}{\sqrt{2\pi} k^2} - i \frac{\sqrt{2\pi} \delta(k)}{2k}.\tag{8} $$ Method 2: A more standard approach found online uses: $$ \mathcal{F}\{xf(x)\} = i \frac{d}{dk} \tilde{f}(k),\tag{9} $$ setting $f(x) = H(x)$ and noting $xH(x) = R(x)$, we have: $$ \mathcal{F}\{R(x)\} = i \frac{d}{dk} \left(\frac{\sqrt{2\pi}}{2} \delta(k) - \frac{i}{\sqrt{2\pi} k}\right) = i \frac{\sqrt{2\pi}}{2} \delta'(k) - \frac{1}{\sqrt{2\pi} k^2}.\tag{10} $$ This clearly differs from the results obtained by the first method. More details on this method can be found at this Math Stack Exchange post.

Remarks: It is noted in this Math Stack Exchange question that: $$ \tilde{H}(k) = \frac{\sqrt{2\pi}}{2} \delta(k) - \frac{i}{\sqrt{2\pi} k},\tag{11} $$ should actually be written as: $$ \tilde{H}(k) = \frac{\sqrt{2\pi}}{2} \delta(k) - P.V. \frac{i}{\sqrt{2\pi} k},\tag{12} $$ I am unsure how this affects the correctness of the above methods, as both omit the P.V. term.

Question: Which method correctly computes the Fourier transform of the ramp function? Where does the incorrect method fail? After much thought, I cannot figure out where the first method goes wrong (as the second approach is generally accepted, I suspect the first method might have an issue).

Your insights would be greatly appreciated.

Answer 1

There is no meaning a priori to $\delta(k)/k$, see e.g. here, and more precisely, the "property of Fourier transform" of dividing by $k$ to integrate that you use in your first answer is not always meaningful for distributions.

Your second method however is right, as long as you define $1/k$ as $\mathrm{pv}(1/k)$ and $1/k^2$ as $\mathrm{fp}(1/k^2)$ (the Hadamard finite part of $1/k^2$), that is the derivative of the principal value of $1/k$.

By the answer I linked, you will however see that the good interpretation of $\delta(k)/k$ indeed tells that your two methods gives actually the same result.